Megan Cardy is one of the lucky ones. Her doctors found a match. Megan was 14 when her leukemia returned. Having already endured chemotherapy, she dreaded the prospect of another long course of treatment. So Megan and her parents decided on a bone-marrow transplant that would provide blood stem cells to enable her body to produce healthy blood. After her doctors spent six months searching the donor lists for stem cells close enough to Megan's own tissue, she got the stem cells she needed — but not from a bone-marrow donor. Instead, the source was umbilical cord blood, a byproduct...